Minimum Height Trees 解答

Question

For an undirected graph with tree characteristics, we can choose any node as the root. The result graph is then a rooted tree. Among all possible rooted trees, those with minimum height are called minimum height trees (MHTs). Given such a graph, write a function to find all the MHTs and return a list of their root labels.

Format
The graph contains n nodes which are labeled from 0 to n - 1. You will be given the number n and a list of undirected edges (each edge is a pair of labels).

You can assume that no duplicate edges will appear in edges. Since all edges are undirected, [0, 1] is the same as [1, 0] and thus will not appear together in edges.

Example 1 :

Input: n = 4, edges = [[1, 0], [1, 2], [1, 3]]

        0
        |
        1
       / \
      2   3 

Output: [1]

Example 2 :

Input: n = 6, edges = [[0, 3], [1, 3], [2, 3], [4, 3], [5, 4]]

     0  1  2
      \ | /
        3
        |
        4
        |
        5 

Output: [3, 4]

Note:

• According to the definition of tree on Wikipedia: “a tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any connected graph without simple cycles is a tree.”
• The height of a rooted tree is the number of edges on the longest downward path between the root and a leaf.

Solution

如果是一条线的话，我们知道选中间的点当root会是minimum height tree。类似的，如果是一堆点，我们通过BFS一层层拨开外面的叶子，那么剩下的就是中心的点。一开始还是需要构建图的adjacency list。

BFS时间复杂度是O(N)

class Solution:
    def findMinHeightTrees(self, n: int, edges: List[List[int]]) -> List[int]:
        if n == 1:
            return [0]
        adjacency_list = [set() for i in range(n)]
        # Build adjacnecy list
        for edge in edges:
            adjacency_list[edge[0]].add(edge[1])
            adjacency_list[edge[1]].add(edge[0])
        # Build leaves list
        leaves = [i for i in range(n) if len(adjacency_list[i]) == 1]
        # BFS
        while n > 2:
            n -= len(leaves)
            new_leaves = []
            while leaves:
                leaf = leaves.pop()
                neighbor = adjacency_list[leaf].pop()
                adjacency_list[neighbor].remove(leaf)
                if len(adjacency_list[neighbor]) == 1:
                    new_leaves.append(neighbor)
            leaves = new_leaves
        return leaves